Zonally propagating wave solutions of Laplace Tidal Equations in a baroclinic ocean of an aqua-planet

De-Leon, Yair; Paldor, Nathan

doi:10.3402/tellusa.v63i2.15783

Cited by 4 publications

(8 citation statements)

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“…In addition, both the Rossby–Haurwitz test case and the proposed one focus on the limit of large gH , but, as the vertical resolution in global‐scale dynamical cores increases, the layers' mean thicknesses decrease and the values of gH used by these two test cases are no longer relevant. For this purpose the solutions obtained by De‐Leon and Paldor (), which are valid in the limit of small gH , can be used to test global‐scale GCMs in the same exact way the solutions obtained by Paldor et al () are used here. In an attempt to include the limit of small values of gH as part of the proposed test case, we have also used the Eulerian spectral dynamical core of the NCAR's model to simulate the solutions obtained by De‐Leon and Paldor () and found that, while the spatial structures of the initial waves were preserved with a comparable accuracy to those found here, their phase speeds were in error of about 40 % .…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…For this purpose the solutions obtained by De‐Leon and Paldor (), which are valid in the limit of small gH , can be used to test global‐scale GCMs in the same exact way the solutions obtained by Paldor et al () are used here. In an attempt to include the limit of small values of gH as part of the proposed test case, we have also used the Eulerian spectral dynamical core of the NCAR's model to simulate the solutions obtained by De‐Leon and Paldor () and found that, while the spatial structures of the initial waves were preserved with a comparable accuracy to those found here, their phase speeds were in error of about 40 % . Since we were unable to pinpoint the origin of this difference and demonstrate that the solutions obtained by De‐Leon and Paldor () can indeed be accurately simulated on a fully operational GCM, we do not include these wave modes in the proposed test case and postpone the treatment of the limit of small gH to future work.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

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A quantitative test case for global‐scale dynamical cores based on analytic wave solutions of the shallow‐water equations

Shamir

Paldor

2016

Quart J Royal Meteoro Soc

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Recently derived analytic wave solutions of the shallow‐water equations (SWEs) on the rotating spherical Earth are employed to construct a test case for hydrostatic dynamical cores of global‐scale general circulation models (GCMs). The proposed test case is more relevant to the SWEs than the frequently used Rossby–Haurwitz test case which is based on wave solutions of the non‐divergent barotropic vorticity equation and not the SWEs. The applicability of the proposed test case to operational GCMs is demonstrated by using the spectral Eulerian dynamical core of the atmospheric component of NCAR's Community Earth System Model to simulate the analytic solutions. An initial slowly propagating Rossby wave and a fast eastward propagating inertia–gravity wave are both accurately simulated for 100 wave periods. In order to quantify the accuracy of the simulations, two error‐measures are suggested which complement the conservation of global energy and, unlike the frequently used L2 error‐measure, provide independent assessments of the errors in the phase speeds and the meridional structures of the simulated waves and are therefore more relevant to periodic wave solutions.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Section: Conclusion and Discussionmentioning

confidence: 99%

A quantitative test case for global‐scale dynamical cores based on analytic wave solutions of the shallow‐water equations

Shamir

Paldor

2016

Quart J Royal Meteoro Soc

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“…In contrast, solutions of system are not known to form a complete set and no relationship can be established between the values of C and the latitudinal structure of V cos ϕ and η . The formulation of an eigenvalue equation associated with waves proved to be highly informative in several set‐ups including the midlatitude β ‐plane (Paldor et al ., ; Paldor and Sigalov, ), an equatorial channel on a sphere (De‐Leon et al ., ) and the entire sphere (De‐Leon and Paldor, ; Paldor et al ., ; Paldor, ). In contrast to these channel problems where the boundary conditions are ‘no normal flow’ at the channel walls, the boundary conditions that solutions of must satisfy on a sphere are their regularity at the singular poles.…”

Section: A Schrödinger Eigenvalue Equation Of the Lrswementioning

confidence: 97%

“…The boundary conditions that V cos ϕ and η have to satisfy are regularity at the (singular) poles, ϕ = ±π/2 where cos ϕ vanishes. As will be discussed in section 5 and as was shown in De‐Leon and Paldor (), Paldor et al . () and Paldor (), it is possible to form an eigenvalue equation associated with this system whose solutions form a complete set and whose eigenvalues are closely related to the number of zero‐crossings of the eigenfunction.…”

Section: A New Perspective On the Degeneracy Of Kelvin Waves On A Rotmentioning

confidence: 97%

“…Recently, Schrödinger eigenvalue problems were formulated for zonally propagating wave solutions of the LRSWE in midlatitudes (Paldor et al ., ; Paldor and Sigalov, ) and approximate Schrödinger eigenvalue problems were also formulated on a sphere (De‐Leon and Paldor, ; Paldor et al ., ; Paldor, ). These formulations provide a clear and concise way of identifying the various waves and modes that solve the LRSWE.…”

Section: Introductionmentioning

confidence: 99%

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Classification of eastward propagating waves on the spherical Earth

Garfinkel

Fouxon

Shamir

et al. 2017

Quart J Royal Meteoro Soc

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Observational evidence for an equatorial non-dispersive mode propagating at the speed of gravity waves is strong, and while the structure and dispersion relation of such a mode can be accurately described by a wave theory on the equatorial β-plane, prior theories on the sphere were unable to find such a mode except for particular asymptotic limits of gravity wave phase speeds and/or certain zonal wave numbers. Here, an ad hoc solution of the linearized rotating shallow-water equations (LRSWE) on a sphere is developed, which propagates eastward with phase speed that nearly equals the speed of gravity waves at all zonal wave numbers. The physical interpretation of this mode in the context of other modes that solve the LRSWE is clarified through numerical calculations and through eigenvalue analysis of a Schrödinger eigenvalue equation that approximates the LRSWE. By comparing the meridional amplitude structure and phase speed of the ad hoc mode with those of the lowest gravity mode on a non-rotating sphere we show that at large zonal wave number the former is a rotation-modified counterpart of the latter. We also find that the dispersion relation of the ad hoc mode is identical to the n = 0 eastward propagating inertia-gravity (EIG0) wave on a rotating sphere which is also nearly non-dispersive, so this solution could be classified as both a Kelvin wave and as the EIG0 wave. This is in contrast to Cartesian coordinates where Kelvin waves are a distinct wave solution that supplements the EIG0 mode. Furthermore, the eigenvalue equation for the meridional velocity on the β-plane can be formally derived as an asymptotic limit (for small (Lamb Number) -1/4 ) of the corresponding second order equation on a sphere, but this expansion is invalid when the phase speed equals that of gravity waves i.e. for Kelvin waves. Various expressions found in the literature for both Kelvin waves and inertia-gravity waves and which are valid only in certain asymptotic limits (e.g. slow and fast rotation) are compared with the expressions found here for the two wave types.

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The Matsuno–Gill model on the sphere

et al. 2023

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We extend the Matsuno–Gill model, originally developed on the equatorial $\beta$ -plane, to the surface of the sphere. While on the $\beta$ -plane the non-dimensional model contains a single parameter, the damping rate $\gamma$ , on a sphere the model contains a second parameter, the rotation rate $\epsilon ^{1/2}$ (Lamb number). By considering the different combinations of damping and rotation, we are able to characterize the solutions over the $(\gamma, \epsilon ^{1/2})$ plane. We find that the $\beta$ -plane approximation is accurate only for fast rotation rates, where gravity waves traverse a fraction of the sphere's diameter in one rotation period. The particular solutions studied by Matsuno and Gill are accurate only for fast rotation and moderate damping rates, where the relaxation time is comparable to the time on which gravity waves traverse the sphere's diameter. Other regions of the parameter space can be described by different approximations, including radiative relaxation, geostrophic, weak temperature gradient and non-rotating approximations. The effect of the additional parameter introduced by the sphere is to alter the eigenmodes of the free system. Thus, unlike the solutions obtained by Matsuno and Gill, where the long-term response to a symmetric forcing consists solely of Kelvin and Rossby waves, the response on the sphere includes other waves as well, depending on the combination of $\gamma$ and $\epsilon ^{1/2}$ . The particular solutions studied by Matsuno and Gill apply to Earth's oceans, while the more general $\beta$ -plane solutions are only somewhat relevant to Earth's troposphere. In Earth's stratosphere, Venus and Titan, only the spherical solutions apply.

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Zonally propagating wave solutions of Laplace Tidal Equations in a baroclinic ocean of an aqua-planet

Cited by 4 publications

References 0 publications

A quantitative test case for global‐scale dynamical cores based on analytic wave solutions of the shallow‐water equations

A quantitative test case for global‐scale dynamical cores based on analytic wave solutions of the shallow‐water equations

Classification of eastward propagating waves on the spherical Earth

The Matsuno–Gill model on the sphere

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